# Cubic reciprocity  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In mathematics, cubic reciprocity refers to various results connecting the solvability of two related cubic equations in modular arithmetic. It is a generalisation of the concept of quadratic reciprocity.

## Algebraic setting

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring E of complex numbers of the form

$z=a+b\,\omega$ where and a and b are integers and

$\omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}$ is a complex cube root of unity.

If $\pi$ is a prime element of E of norm P and $\alpha$ is an element coprime to $\pi$ , we define the cubic residue symbol $\left({\frac {\alpha }{\pi }}\right)_{3}$ to be the cube root of unity (power of $\omega$ ) satisfying

$\alpha ^{(P-1)/3}\equiv \left({\frac {\alpha }{\pi }}\right)_{3}\mod \pi$ We further define a primary prime to be one which is congruent to -1 modulo 3. Then for distinct primary primes $\pi$ and $\theta$ the law of cubic reciprocity is simply

$\left({\frac {\pi }{\theta }}\right)_{3}=\left({\frac {\theta }{\pi }}\right)_{3}$ with the supplementary laws for the units and for the prime $1-\omega$ of norm 3 that if $\pi =-1+3(m+n\omega )$ then

$\left({\frac {\omega }{\pi }}\right)_{3}=\omega ^{m+n}$ $\left({\frac {1-\omega }{\pi }}\right)_{3}=\omega ^{2m}$ 